During the very early years of Islam, what were relations like between the new faith and Christians in particular? Was it akin to a "reformist" movement pulling people with it into a new faith?

This is a point of quite a lot of debate and as far as I'm aware there is no consensus on the matter. It also depends a good deal on what you mean by "the very early years of Islam."

One of the unresolved questions here is at what point in Muhammad's mission did he conceive of his revelations as being a separate and unique revelation from previous ones (i.e. Christian or Jewish revelation)?

As an aside, it's worth repeating here that insofar as there is consensus it's that it would be impossible to differentiate these influences as being predominantly Christian or Jewish but should perhaps be better understood as Abrahamic monotheistic.

But in any event, certainly by the later Medinan period of Muhammad's life it seems that seems that he came to understand his revelation as new and separate, as in Quran 5:3 (one of the later revelations of Muhammad's life) which reads: "This day have I perfected your religion for you and completed My favour unto you, and have chosen for you as religion al-Islam."

But there has been much speculation and debate about the extent to which this may yet have been viewed as a kind of syncretist or reformist religion or whether it shifted from being so at some point to being more radically a unique religious entity. As far as I'm aware this debate in its modern academic form goes back at least as far as Richard Bell writing in the 1920s (in its Christian apologetic form it obviously goes back much further as /u/shlin28's comment on 'Islam as heresy' indicates.)

It's worth quoting Bell at length because as far as I'm aware his idea here has never really been refuted, so, from Bell's The Origin of Islam in its Christian Environment he says:

We have seen that Muhammad's first enterprise was the reform of religion in Mecca. There was only One God, who had claims upon the grateful worship of man. He knew that the worship of the One God prevailed all round Arabia, and especially in that great Roman Empire which must have seemed to an intelligent Arab, dissatisfied with the condition of his own people, the seat of enlightenment, culture, and civilisation. What more natural than that he should turn thither for information as to what the worship of the one God involved? May we suppose him to have seen Christian services on some youthful visit to Syria, or should we suppose him simply to have made inquiries when his own plan began to take shape in his mind? In either case he would come to realise that the worship of the one God included stated services or prayers. Thus he instituted the salat, the formal prayer or divine service. The word which denotes it is a borrowed Christian word. It is not quite clear that the salat was practised from the very beginning, but it is certain that it was a very early institution of Islam, though the five prayers at stated times of the day were not regulated until much later. In any case the religion of the One God implied the possession of a Holy Scripture. Even if he had never been out of Mecca Muhammad could not have been ignorant of the possession by Christians of a Book which was believed to have been directly revealed. For as we have seen that was common knowledge among the Arabs. What that Book contained was not so generally known and was certainly in my opinion not known to Muhammad. But the existence of the Book itself he must have known about.

How Muhammad came to believe that he was commissioned to produce such a Scripture or Book of Revelation for his own countrymen is difficult for us to understand. Tradition, as I have said, is a very unsafe guide. The Qur'an does contain one or two references to a vision, or perhaps two visions, in passages which are fairly early, but certainly not among the very earliest:

By the declining star, Your comrade is not astray, nor does he err, Nor does he speak of his own desire. It is nothing but a revelation revealed Taught by one strong in power Of definite form. With equal poise he stood While he was on the high horizon. Then he drew near and came down Until he was two bow-lengths off or nearer, And revealed to his servant what he revealed. The heart does not falsify what it saw. Do you dispute with it as to what it sees? He, saw him too when he came down a second time By the lote-tree at the boundary Near the garden of the dwelling, When the lote-tree was enveloped by what enveloped it. The eye turned not aside nor caused illusion. Verily he saw one of the great signs of his Lord. (liii. vv. 1-18.)

But if such a vision lay at the root of his prophetic work, he made curiously little of it, only referring to it in one other passage ; and in Surah xvii. v. 62 we find the curious statement: "We appointed the vision which we showed you only as a test for men, likewise the tree cursed in the Qur'an. We terrify them but they only increase in crass rebellion."

Perhaps we are too ready to follow the Moslem commentators and traditionists in assuming that the idea of the Book sent down from Heaven and revealed to him as occasion required by the intermediary of the Angel Gabriel was in Muhammad's mind from the very beginning. There is no real evidence of that in the Qur'an itself. The word Book (kitab) is used in early Meccan portions of the Qur'an in an entirely different sense from that of the heavenly archetype of the revelation. On the Judgement Day the Book will be produced. Every soul will be brought face to face with its book, which will be put in its right hand or its left, according as it has done well or ill. This is evidently something like the idea of the Book kept in Heaven by the Recording Angel. The idea of the Book as the heavenly archetype of the Qur'an belongs to a later stage. There are indeed frequent references to revelation in the very earliest passages of the Qur'an, but in practically every case it is possible to interpret these as references to the Revelation already in the hands of previous Monotheists. When, for instance, we read in Surah lxxx. vv. 11-15—a passage which has not much connection with what precedes or follows: Nay, it is a Warning, Take heed of it who will, In leaves, honoured, Exalted, kept pure By the hands of scribes worthy and pious, it is only dogmatic interpretation which finds there a reference to the heavenly archetype of the Qur'an. It is much more natural to take it as a reference to the Bible so carefully preserved by Christians or by Jews. The statement (lxxxv. vv. 21, 22):

Very it is a glorious Qur'an in a preserved Tablet.

may, and if it is early, I think should be, similarly interpreted, especially as by a mere change of vowel it can be made to read "preserved in a tablet". When we find references in equally early passages to as-suhuf al-ula, the first or former pages, and to suhuf Ibrahim wa Musa, "the pages of Abraham and Moses", or to the "pages of Moses" alone, it is clear that what is in Muhammad's mind is primarily the Revelation as in the hands of men. That is what is so carefully preserved. He has been making inquiries about that revelation, and has learned so much about it. That his sources of information were very restricted and imperfect is shown by the references to "the pages of Abraham".

We shall see later at what stage the idea of the Book given him from Heaven took definite shape in his mind. His original conception of his messages was probably a much more modest one. My own impression is that he began to produce these geryane or oracles without any very exalted idea of his own prophetic office, and that his prophetic consciousness gradually developed as his mission proceeded, the process being aided by his preoccupation with former prophets. Probably his purpose was to reproduce in Arabic form the gist of what had been revealed to other peoples.

This has been expanded and altered, in one prominent example, in Patricia Crone and Michael Cook's Hagarism which renders Islam into a Judeo-Hagaritic (i.e of the people descended from Abraham's slave Hagar) religion which differentiated itself from Judaism primarily by adopting Christ as the Messiah, but not as the savior. The authors later repudiated the thesis, as I understand it, but this search for a better understanding of what exactly Islam was for the early believers has continued in, for instance, Fred Donner's Muhammad and the Believers: At the Origins of Islam which posits that what Muhammad did was to form a community (umma) of believers (mua'minin) which was the primary identity of a group that also un-controversially included Christians and Muslims.

For instance, Donner writes:

"We should also take note here of a comment made by the Arme-nian bishop Sebeos, writing in the 660s, whose chronicle provides one of the earliest extant descriptions of the Believers and their ac- tions. Among other things, he notes that the Believers' first governor of Jerusalem was a Jew. Imputation of close ties to Judaism was a favorite polemical technique used by Christian authors of this pe- riod to discredit their opponents, so this claim needs to be handled with caution; but if true, it provides further evidence of the confessional, open character of the Believers' movement. It is usually claimed that, with the arrival of the "Muslims," Christians and Jews were either reduced to subject status as "protected peoples" (ahl al- dhimma), who paid tax but formed a distinctly lower level in society, or converted to Islam and incorporated into the Arabian Muslims as clients (mawali). But the passages from Bar Penkaye, IshoVahb, and Sebeos noted above, and a few others like them, suggest that some Christians and Jews may have been fully integrated, as such, into the early community of Believers; conversion was not at issue, because as monotheists they did not need to "convert" to anything in order to become active participants in the community, even in areas outside Arabia. “ - Muhammad and the Believers: At the Origins of Islam - Fred Donner, 114

TBC